| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | isperp.p | . . . 4
⊢ 𝑃 = (Base‘𝐺) | 
| 2 |  | isperp.d | . . . 4
⊢  − =
(dist‘𝐺) | 
| 3 |  | isperp.i | . . . 4
⊢ 𝐼 = (Itv‘𝐺) | 
| 4 |  | isperp.l | . . . 4
⊢ 𝐿 = (LineG‘𝐺) | 
| 5 |  | eqid 2737 | . . . 4
⊢
(pInvG‘𝐺) =
(pInvG‘𝐺) | 
| 6 |  | isperp.g | . . . . 5
⊢ (𝜑 → 𝐺 ∈ TarskiG) | 
| 7 | 6 | adantr 480 | . . . 4
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝐺 ∈ TarskiG) | 
| 8 |  | ragperp.b | . . . . . 6
⊢ (𝜑 → 𝐵 ∈ ran 𝐿) | 
| 9 | 8 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝐵 ∈ ran 𝐿) | 
| 10 |  | simprr 773 | . . . . 5
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑣 ∈ 𝐵) | 
| 11 | 1, 4, 3, 7, 9, 10 | tglnpt 28557 | . . . 4
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑣 ∈ 𝑃) | 
| 12 |  | isperp.a | . . . . . 6
⊢ (𝜑 → 𝐴 ∈ ran 𝐿) | 
| 13 | 12 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝐴 ∈ ran 𝐿) | 
| 14 |  | ragperp.x | . . . . . . 7
⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) | 
| 15 | 14 | elin1d 4204 | . . . . . 6
⊢ (𝜑 → 𝑋 ∈ 𝐴) | 
| 16 | 15 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑋 ∈ 𝐴) | 
| 17 | 1, 4, 3, 7, 13, 16 | tglnpt 28557 | . . . 4
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑋 ∈ 𝑃) | 
| 18 |  | simprl 771 | . . . . 5
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑢 ∈ 𝐴) | 
| 19 | 1, 4, 3, 7, 13, 18 | tglnpt 28557 | . . . 4
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑢 ∈ 𝑃) | 
| 20 |  | ragperp.v | . . . . . . 7
⊢ (𝜑 → 𝑉 ∈ 𝐵) | 
| 21 | 20 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑉 ∈ 𝐵) | 
| 22 | 1, 4, 3, 7, 9, 21 | tglnpt 28557 | . . . . 5
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑉 ∈ 𝑃) | 
| 23 |  | ragperp.u | . . . . . . . . 9
⊢ (𝜑 → 𝑈 ∈ 𝐴) | 
| 24 | 23 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑈 ∈ 𝐴) | 
| 25 | 1, 4, 3, 7, 13, 24 | tglnpt 28557 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑈 ∈ 𝑃) | 
| 26 |  | ragperp.r | . . . . . . . 8
⊢ (𝜑 → 〈“𝑈𝑋𝑉”〉 ∈ (∟G‘𝐺)) | 
| 27 | 26 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 〈“𝑈𝑋𝑉”〉 ∈ (∟G‘𝐺)) | 
| 28 |  | ragperp.1 | . . . . . . . 8
⊢ (𝜑 → 𝑈 ≠ 𝑋) | 
| 29 | 28 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑈 ≠ 𝑋) | 
| 30 | 23 | ad2antrr 726 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑈 ∈ 𝐴) | 
| 31 | 6 | ad2antrr 726 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝐺 ∈ TarskiG) | 
| 32 | 17 | adantr 480 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑋 ∈ 𝑃) | 
| 33 | 19 | adantr 480 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑢 ∈ 𝑃) | 
| 34 |  | simpr 484 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → ¬ 𝑋 = 𝑢) | 
| 35 | 34 | neqned 2947 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑋 ≠ 𝑢) | 
| 36 | 12 | ad2antrr 726 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝐴 ∈ ran 𝐿) | 
| 37 | 15 | ad2antrr 726 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑋 ∈ 𝐴) | 
| 38 | 18 | adantr 480 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑢 ∈ 𝐴) | 
| 39 | 1, 3, 4, 31, 32, 33, 35, 35, 36, 37, 38 | tglinethru 28644 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝐴 = (𝑋𝐿𝑢)) | 
| 40 | 30, 39 | eleqtrd 2843 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑢) → 𝑈 ∈ (𝑋𝐿𝑢)) | 
| 41 | 40 | ex 412 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (¬ 𝑋 = 𝑢 → 𝑈 ∈ (𝑋𝐿𝑢))) | 
| 42 | 41 | orrd 864 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (𝑋 = 𝑢 ∨ 𝑈 ∈ (𝑋𝐿𝑢))) | 
| 43 | 42 | orcomd 872 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (𝑈 ∈ (𝑋𝐿𝑢) ∨ 𝑋 = 𝑢)) | 
| 44 | 1, 2, 3, 4, 5, 7, 25, 17, 22, 19, 27, 29, 43 | ragcol 28707 | . . . . . 6
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 〈“𝑢𝑋𝑉”〉 ∈ (∟G‘𝐺)) | 
| 45 | 1, 2, 3, 4, 5, 7, 19, 17, 22, 44 | ragcom 28706 | . . . . 5
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 〈“𝑉𝑋𝑢”〉 ∈ (∟G‘𝐺)) | 
| 46 |  | ragperp.2 | . . . . . 6
⊢ (𝜑 → 𝑉 ≠ 𝑋) | 
| 47 | 46 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 𝑉 ≠ 𝑋) | 
| 48 | 20 | ad2antrr 726 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑉 ∈ 𝐵) | 
| 49 | 6 | ad2antrr 726 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝐺 ∈ TarskiG) | 
| 50 | 17 | adantr 480 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑋 ∈ 𝑃) | 
| 51 | 11 | adantr 480 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑣 ∈ 𝑃) | 
| 52 |  | simpr 484 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → ¬ 𝑋 = 𝑣) | 
| 53 | 52 | neqned 2947 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑋 ≠ 𝑣) | 
| 54 | 8 | ad2antrr 726 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝐵 ∈ ran 𝐿) | 
| 55 | 14 | elin2d 4205 | . . . . . . . . . . 11
⊢ (𝜑 → 𝑋 ∈ 𝐵) | 
| 56 | 55 | ad2antrr 726 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑋 ∈ 𝐵) | 
| 57 | 10 | adantr 480 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑣 ∈ 𝐵) | 
| 58 | 1, 3, 4, 49, 50, 51, 53, 53, 54, 56, 57 | tglinethru 28644 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝐵 = (𝑋𝐿𝑣)) | 
| 59 | 48, 58 | eleqtrd 2843 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) ∧ ¬ 𝑋 = 𝑣) → 𝑉 ∈ (𝑋𝐿𝑣)) | 
| 60 | 59 | ex 412 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (¬ 𝑋 = 𝑣 → 𝑉 ∈ (𝑋𝐿𝑣))) | 
| 61 | 60 | orrd 864 | . . . . . 6
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (𝑋 = 𝑣 ∨ 𝑉 ∈ (𝑋𝐿𝑣))) | 
| 62 | 61 | orcomd 872 | . . . . 5
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → (𝑉 ∈ (𝑋𝐿𝑣) ∨ 𝑋 = 𝑣)) | 
| 63 | 1, 2, 3, 4, 5, 7, 22, 17, 19, 11, 45, 47, 62 | ragcol 28707 | . . . 4
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 〈“𝑣𝑋𝑢”〉 ∈ (∟G‘𝐺)) | 
| 64 | 1, 2, 3, 4, 5, 7, 11, 17, 19, 63 | ragcom 28706 | . . 3
⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐵)) → 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺)) | 
| 65 | 64 | ralrimivva 3202 | . 2
⊢ (𝜑 → ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺)) | 
| 66 | 1, 2, 3, 4, 6, 12,
8, 14 | isperp2 28723 | . 2
⊢ (𝜑 → (𝐴(⟂G‘𝐺)𝐵 ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐵 〈“𝑢𝑋𝑣”〉 ∈ (∟G‘𝐺))) | 
| 67 | 65, 66 | mpbird 257 | 1
⊢ (𝜑 → 𝐴(⟂G‘𝐺)𝐵) |